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Game theory
(Sections 17.5-17.6)
Game theory
• Game theory deals with systems of interacting
agents where the outcome for an agent depends
on the actions of all the other agents
– Applied in sociology, politics, economics, biology,
and, of course, AI
• Agent design: determining the best strategy for
a rational agent in a given game
• Mechanism design: how to set the rules of the
game to ensure a desirable outcome
http://www.economist.com/node/21527025
Simultaneous single-move games
• Players must choose their actions at the same time, without
knowing what the others will do
– Form of partial observability
Normal form representation:
Player 1
Player 2
0,0
1,-1
-1,1
-1,1
0,0
1,-1
1,-1
-1,1
0,0
Payoff matrix
(Player 1’s utility is listed first)
Is this a zero-sum game?
Prisoner’s dilemma
• Two criminals have been
arrested and the police visit
them separately
• If one player testifies against
the other and the other refuses,
the one who testified goes free
and the one who refused gets a
10-year sentence
• If both players testify against
each other, they each get a
5-year sentence
• If both refuse to testify, they
each get a 1-year sentence
Alice:
Testify
Alice:
Refuse
Bob:
Testify
-5,-5
-10,0
Bob:
Refuse
0,-10
-1,-1
Prisoner’s dilemma
• Alice’s reasoning:
– Suppose Bob testifies. Then I get
5 years if I testify and 10 years if
I refuse. So I should testify.
– Suppose Bob refuses. Then I go
free if I testify, and get 1 year if
I refuse. So I should testify.
• Dominant strategy: A strategy
whose outcome is better for the
player regardless of the strategy
chosen by the other player
Alice:
Testify
Alice:
Refuse
Bob:
Testify
-5,-5
-10,0
Bob:
Refuse
0,-10
-1,-1
Prisoner’s dilemma
• Nash equilibrium: A pair of
strategies such that no player
can get a bigger payoff by
switching strategies, provided
the other player sticks with the
same strategy
– (Testify, testify) is a dominant
strategy equilibrium
• Pareto optimal outcome: It is
impossible to make one of the
players better off without making
another one worse off
• In a non-zero-sum game, a Nash
equilibrium is not necessarily
Pareto optimal!
Alice:
Testify
Alice:
Refuse
Bob:
Testify
-5,-5
-10,0
Bob:
Refuse
0,-10
-1,-1
Recall: Multi-player, non-zero-sum game
4,3,2
4,3,2
4,3,2
1,5,2
7,4,1
1,5,2
7,7,1
Prisoner’s dilemma in real life
•
•
•
•
•
Price war
Defect
Arms race
Cooperate
Steroid use
Diner’s dilemma
Collective action in politics
Defect
Cooperate
Lose – lose
Lose big –
win big
Win big –
lose big
Win – win
http://en.wikipedia.org/wiki/Prisoner’s_dilemma
Collective action in economics
Is there any way to get
a better answer?
• Superrationality
– Assume that the answer to a symmetric problem will
be the same for both players
– Maximize the payoff to each player while considering
only identical strategies
– Not a conventional model in game theory
• Repeated games
– If the number of rounds is fixed and known in
advance, the equilibrium strategy is still to defect
– If the number of rounds is unknown, cooperation may
become an equilibrium strategy
Stag hunt
Hunter 1: Hunter 1:
Stag
Hare
Hunter 2:
Stag
2,2
1,0
Hunter 2:
Hare
0,1
1,1
• Is there a dominant strategy for either player?
• Is there a Nash equilibrium?
– (Stag, stag) and (hare, hare)
• Model for cooperative activity
Prisoner’s dilemma
vs. stag hunt
Stag hunt
Prisoner’ dilemma
Cooperate
Defect
Cooperate
Win – win
Win big –
lose big
Defect
Lose big –
win big
Lose – lose
Players can gain by
defecting unilaterally
Cooperate
Defect
Cooperate
Win big –
win big
Win – lose
Defect
Lose – win
Win – win
Players lose by
defecting unilaterally
Game of Chicken
Player 1
S
Player 2
Straight
Chicken
Chicken
Straight
C
S -10, -10
-1, 1
C
0, 0
1, -1
• Is there a dominant strategy for either player?
• Is there a Nash equilibrium?
(Straight, chicken) or (chicken, straight)
• Anti-coordination game: it is mutually beneficial for the
two players to choose different strategies
– Model of escalated conflict in humans and animals
(hawk-dove game)
• How are the players to decide what to do?
– Pre-commitment or threats
– Different roles: the “hawk” is the territory owner and the “dove” is
the intruder, or vice versa
http://en.wikipedia.org/wiki/Game_of_chicken
Mixed strategy equilibria
Player 1
S
Player 2
Straight
Chicken
Chicken
Straight
C
S -10, -10
-1, 1
C
0, 0
1, -1
• Mixed strategy: a player chooses between the moves
according to a probability distribution
• Suppose each player chooses S with probability 1/10.
Is that a Nash equilibrium?
• Consider payoffs to P1 while keeping P2’s strategy fixed
–
–
–
–
The payoff of P1 choosing S is (1/10)(–10) + (9/10)1 = –1/10
The payoff of P1 choosing C is (1/10)(–1) + (9/10)0 = –1/10
Can P1 change their strategy to get a better payoff?
Same reasoning applies to P2
Finding mixed strategy equilibria
P1: Choose S
with prob. p
P1: Choose C
with prob. 1-p
P2: Choose S
with prob. q
-10, -10
-1, 1
P2: Choose C
with prob. 1-q
1, -1
0, 0
• Expected payoffs for P1 given P2’s strategy:
P1 chooses S: q(–10) +(1–q)1 = –11q + 1
P1 chooses C: q(–1) + (1–q)0 = –q
• In order for P2’s strategy to be part of a Nash
equilibrium, P1 has to be indifferent between its two
actions:
–11q + 1 = –q or q = 1/10
Similarly, p = 1/10
Ultimatum game
• Alice and Bob are given a sum of money S to divide
–
–
–
–
Alice picks A, the amount she wants to keep for herself
Bob picks B, the smallest amount of money he is willing to accept
If S – A  B, Alice gets A and Bob gets S – A
If S – A < B, both players get nothing
• What is the Nash equilibrium?
– Alice offers Bob the smallest amount of money he will accept:
S–A=B
– Alice and Bob both want to keep the full amount: A = S, B = S
(both players get nothing)
• How would humans behave in this game?
– If Bob perceives Alice’s offer as unfair, Bob will be likely to refuse
– Is this rational?
• Maybe Bob gets some positive utility for “punishing” Alice?
http://en.wikipedia.org/wiki/Ultimatum_game
Existence of Nash equilibria
• Any game with a finite set of actions has at least
one Nash equilibrium (which may be a mixedstrategy equilibrium)
• If a player has a dominant strategy, there exists a
Nash equilibrium in which the player plays that
strategy and the other player plays the best
response to that strategy
• If both players have strictly dominant strategies,
there exists a Nash equilibrium in which they play
those strategies
Computing Nash equilibria
• For a two-player zero-sum game, simple linear
programming problem
• For non-zero-sum games, the algorithm has worstcase running time that is exponential in the number
of actions
• For more than two players, and for sequential
games, things get pretty hairy
Nash equilibria and rational
decisions
• If a game has a unique Nash equilibrium, it will
be adopted if each player
–
–
–
–
is rational and the payoff matrix is accurate
doesn’t make mistakes in execution
is capable of computing the Nash equilibrium
believes that a deviation in strategy on their part will
not cause the other players to deviate
– there is common knowledge that all players meet
these conditions
http://en.wikipedia.org/wiki/Nash_equilibrium
Mechanism design
(inverse game theory)
• Assuming that agents pick rational strategies, how
should we design the game to achieve a socially
desirable outcome?
• We have multiple agents and a center that
collects their choices and determines the outcome
Auctions
• Goals
– Maximize revenue to the seller
– Efficiency: make sure the buyer who values the goods
the most gets them
– Minimize transaction costs for buyer and sellers
Ascending-bid auction
• What’s the optimal strategy for a buyer?
– Bid until the current bid value exceeds your private value
• Usually revenue-maximizing and efficient, unless
the reserve price is set too low or too high
• Disadvantages
– Collusion
– Lack of competition
– Has high communication costs
Sealed-bid auction
• Each buyer makes a single bid and communicates it to the
auctioneer, but not to the other bidders
– Simpler communication
– More complicated decision-making: the strategy of a buyer depends on
what they believe about the other buyers
– Not necessarily efficient
• Sealed-bid second-price auction: the winner pays the price
of the second-highest bid
–
–
–
–
–
Let V be your private value and B be the highest bid by any other buyer
If V > B, your optimal strategy is to bid above B – in particular, bid V
If V < B, your optimal strategy is to bid below B – in particular, bid V
Therefore, your dominant strategy is to bid V
This is a truth revealing mechanism
Dollar auction
• A dollar bill is auctioned off to the highest bidder, but the
second-highest bidder has to pay the amount of his last bid
–
–
–
–
–
Player 1 bids 1 cent
Player 2 bids 2 cents
…
Player 2 bids 98 cents
Player 1 bids 99 cents
• If Player 2 passes, he loses 98 cents, if he bids $1, he might still come out even
– So Player 2 bids $1
• Now, if Player 1 passes, he loses 99 cents, if he bids $1.01, he only loses 1 cent
– …
• What went wrong?
– When figuring out the expected utility of a bid, a rational player
should take into account the future course of the game
• What if Player 1 starts by bidding 99 cents?
Regulatory mechanism design:
Tragedy of the commons
• States want to set their policies for controlling emissions
– Each state can reduce their emissions at a cost of -10
or continue to pollute at a cost of -5
– If a state decides to pollute, -1 is added to the utility of every
other state
• What is the dominant strategy for each state?
– Continue to pollute
– Each state incurs cost of -5-49 = -54
– If they all decided to deal with emissions, they would incur a cost
of only -10 each
• Mechanism for fixing the problem:
– Tax each state by the total amount by which they reduce the
global utility (externality cost)
– This way, continuing to pollute would now cost -54
Review: Game theory
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•
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•
•
•
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Normal form representation of a game
Dominant strategies
Nash equilibria
Pareto optimal outcomes
Pure strategies and mixed strategies
Examples of games
Mechanism design
– Auctions: ascending bid, sealed bid, sealed bid
second-price, “dollar auction”